## Abstract

In this chapter, we investigate a magnetic line source diffraction problem concerned with a step. To study the diffraction problem in lossy medium, we follow the Wiener-Hopf technique and steepest decent method to solve it for impedance step. By equating the impedances of the step to zero, the solution reduces for magnetic line source diffraction by PEC step. Then we transform the obtained solution for PEMC step by using duality transformation. Perfect electromagnetic conductor (PEMC) theory is novel idea developed by Lindell and Sihvola. This media is interlinked with two conductors namely perfect electric conductor (PEC) and perfect magnetic conductor (PMC). Both PEC and PMC are the limiting cases of perfect electromagnetic conductor (PEMC). We study the magnetic line source diffraction by PEMC step placed in different soils (i) gravel sand (ii) sand and (iii) clay. By using the permittivity, permeability and conductivity of these lossy mediums we predict the loss effect on the diffracted field. Such kind of study is very useful in antenna and wave propagation for subsurface targets and to investigate antenna radiation patterns.

### Keywords

- Wiener-Hopf technique
- Fourier transform
- Green function
- impedance
- diffraction
- line source
- step
- PEMC
- PMC
- PEC
- Lossy medium
- permeability
- conductivity
- permittivity

## 1. Introduction

In this chapter, we have studied the diffraction problem precisely and investigated the magnetic line source diffraction by a perfect electromagnetic conductor (PEMC) step [1, 2, 3] for the lossy medium. PEMC step is assumed to be placed in lossy medium. Discontinuity in diffraction theory is relevant to many engineering applications. The physical significance of the step problem regarding engineering application is due to the fact that it is used in many electronic devices such as solder pad which have many applications in them which are interconnected through a step like circuit, microwave oven etc. This configuration is significant for predicting the scattering caused by an abrupt change in the material as well as in the geometrical properties of a surface. This problem is concerned with the diffraction of plane, cylindrical and surface waves by different impedance step discontinuities, such as step discontinuities made of plasmonic materials. Specially diffraction by a step in a perfectly conducting plane makes a canonical problem for the geometrical theory of diffraction (GTD) analysis of scattering by metallic tapes on paneled compact range reflectors [4]. The scattering of surface waves by the junction of two semi-infinite planes joined together by a step was first introduced by Johansen [5] in the case where both the half planes and the step are characterized by the same surface impedances. This problem is solved by using Wiener-Hopf technique, Green function and steepest descent method. The diffracted far field is investigated by the method of steepest descent. Some of the other researchers like Büyükaksoy and Birbir, Büyükaksoy and Tayyar, Büyükaksoy and Tayyar, Aksoy and Alkumru [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] have been investigated the scattering problems which can also be considered for the diffraction of plane, cylindrical and surface waves by different impedance step discontinuities, such as step discontinuities made of plasmonic materials.

The importance of present work stems from the facts that: (a) the scattering properties of a surface are functions of both its geometrical and material properties. (b) The edge scattering by dihedral structures whose surfaces can be modeled by the impedance boundary condition has been the focus of attention of many scientists for both acoustic and electromagnetic waves [17]. (c) The step geometry constitutes a canonical problem for scattering because a step geometry is used as an interconnection circuit of many electronic devices such as solder pad, microwave oven and frequency selective surface etc. [18]. A diffraction problem due to a magnetic line source is considered as better substitute than the plane waves. It is pertinent to mention here that the problem of diffraction of plane or line source diffraction of electromagnetic waves from a step is both mathematically difficult and physically important because the solution of the problem involves determination of

It is clear that in the case of the line source incidence, the results of plane wave diffraction by impedance step are modified by a multiplicative factor of the form

and

PEMC behaves as an example of an ideal boundary. As a check, we obtain the PMC and PEC boundary conditions as the two limiting case of PEMC:

and

This medium is characterized by a scalar parameter M known as admittance of the surface. PEMC is a generalization of both perfect electric conductor (PEC) and perfect magnetic conductor (PMC) media. Therefore, the medium is known as PEMC. Defining a certain class of duality transformations, this medium corresponds to PEC or PMC media. PEMC medium allows some nonzero fields, it rejects electromagnetic field propagation and acts as a boundary to electromagnetic waves just like the PEC and PMC media. Denoting the unit normal between air and PEMC by, from the continuity of tangential component, the electric field

It is also continuous through the PEMC air interface, because this vanishes in the PEMC medium, and the boundary condition becomes

Similarly, the normal component of the field satisfies

and is continuous across the boundary for

Because the normal component of the Poynting vector at the PEMC boundary vanishes and is nonreciprocal, except in the PMC and PEC limiting cases

We extend the problem reported by [3] for the lossy background medium. Several canonical objects in lossy media has been investigated over the years by many authors [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56] by applying approximate values of electric conductivity and dielectric constants of various materials. The concept of subsurface scattering of EM waves for detecting the cavities and targets buried in soil has important applications in the areas of nonproliferation of weapons, environmental monitoring, hazardous-waste site location and assessment, and even archeology. To have information about this potential, it is first essential to understand the behavior of the soil by applying EM wave, and how the targets within the soil give response. We analyze the response of the soil to an EM wave by using complex dielectric permittivity of the soil in finding radar range resolution. This leads to a concept of an optimum frequency and bandwidth for imaging in a particular soil. The radar cross section of several canonical objects in lossy media is derived, and examples are given for several objects like scattering of buried PEC sphere, PEC cylinder, and PEC plate [44] and similarly scattering by PEMC plate [54], PEMC strip [55] and PEMC cylinder [56]. Furthermore, we can study the diffraction by PEC and PEMC half plane [39] and step is also made by semi-infinite half planes with a step height a, so they can also be investigated for diffraction by using the electric parameters of soils. Also characteristics of radar cross section can be further studied with different objects for PEC, PMC and PEMC cases in lossy medium. In addition to RCS of various PEC, PMC and PEMC objects [59] in lossy medium can also be investigated in future.

The objective of this chapter is to determine the diffracted field by PEMC step excited by a line source in lossy medium and to investigate surface and borehole techniques for detecting and mapping subsurface cavities, targets and to evaluate the results of surface and borehole radar probings performed at the test sites. Detection of subsurface cavities is concerned with ground-probing radar. A number of factors that control the velocity, absorption and attenuation characteristics of a radar wave and plane EM wave propagating through a dielectric as well as lossy medium like the earth. The imaging of objects buried in soil has potentially valuable applications in many diverse areas, such as nonproliferation of weapons, environmental monitoring, hazardous-waste site location and assessment, and even archeology. We study the magnetic line source diffraction by PEMC step placed in different soils (i) gravel sand (ii) sand and (iii) clay. By using the approximate value of permittivity, permeability and conductivity of these lossy mediums, we predict and analyze the loss effect on the diffracted field.

## 2. Mathematical model

Consider the diffraction due to a magnetic line source located at

The time dependence

The Helmholtz equation concerned with the diffraction problem is given below

subject to the boundary conditions at two half planes and a step given by:

## 3. Boundary conditions

and

with continuity equations:

and

where

Here,

and

The solution of the incident field and reflected field from [11] can be written as

where

The diffracted field

and

and

where

## 4. Fourier transform

Taking Fourier transform of the Eq. (10) such that:

and

where

and Eq. (10) reduce to

where

Apply half range Fourier transforms to the Eq. (11)

where

Fourier transforms of the Eqs. (12)–(16) can be written as

and

and

where

The solution of Eq. (17) satisfying the radiation condition for

where

where prime denotes differentiation with respect to

where

From the Eqs (22), (23), (26) and (27), we obtain the following Wiener-Hopf functional equations

The corrected solution of Wiener-Hopf equation [2] in case of line source can be expressed as

where

and

## 5. Far zone solution

The unknown constant

where

## 6. Magnetic line source diffraction by PEC step

The asymptotic solution for the field diffracted by perfect electric conductor (PEC) step is obtained by equating

where

and

such that

Next we transform magnetic line source diffracted field from PEC to PEMC step under the duality transformations in the [21]. The field diffracted by perfectly electric conducting (PEC) step can be transformed.

## 7. Magnetic line source diffraction by PEMC step

We obtain a solution for magnetic line source diffraction by PEMC step by applying a transformation introduced by Lindell and Sihvola, that is known as duality transformation [21]:

where

Moreover, the transformation

gives

where

and

where

## 8. Magnetic line source diffraction by PEMC step in lossy medium

When we study magnetic line source diffraction by PEMC step in lossy medium, we just replace free-space wave number k by

and

where

where

where

## 9. Results and discussion

In this section we discuss some graphical results which have been presented in [3] to predict the effects of the admittance parameters

## 10. Conclusion

It is concluded that the both coupled electric and magnetic fields excitation can be observed analytically for PEMC theory that leads to a most general case for the magnetic line source diffraction by step embedded in lossy medium. The lossy medium is assumed to be made of three different soils (i) gravel sand, (ii) sand and (iii) clay. We see from their respective electric parameters namely permittivity, permeability and conductivity, as the loss increases the amplitude of the diffracted field decreases. By applying this technique to detect the subsurface targets, we can use various soil models. Further, in this chapter at a time we studied diffraction by step using PEMC theory and loss effect on the field patterns. Here, we can predict the behavior of the fields diffracted by magnetic line source. This is the most general solution and is more useful rather a plane wave solution. In far zone, we can obtain a solution for the diffraction of a plane wave by PEMC step placed in lossy medium under the condition

## Acknowledgments

The author Dr. Saeed Ahmed acknowledges the financial support from the Department of Earth Sciences, Quaid-i-Azam University, Islamabad, Pakistan, during the Post Doctoral studies for the year (20 January, 2017–19 January, 2018).